Virtual metrology systems and methods for using feedforward critical dimension data to predict other critical dimensions of a wafer

ABSTRACT

A controller includes a memory that stores a first model corresponding to a first critical dimension of a substrate processed by a substrate processing system and a second model corresponding to a second critical dimension of the substrate. The second model includes a predicted relationship between the first critical dimension and the second critical dimension. A critical dimension prediction module calculates a first prediction of the first critical dimension of the substrate using the first model, provides the first prediction of the first critical dimension as an input to the second model, and calculates and outputs a second prediction of the second critical dimension of the substrate using the second model.

FIELD

The present disclosure relates to substrate processing systems, and moreparticularly to predicting critical dimensions of a substrate usingvirtual metrology.

BACKGROUND

The background description provided here is for the purpose of generallypresenting the context of the disclosure. Work of the presently namedinventors, to the extent it is described in this background section, aswell as aspects of the description that may not otherwise qualify asprior art at the time of filing, are neither expressly nor impliedlyadmitted as prior art against the present disclosure.

Substrate processing systems may be used to perform etching, deposition,and/or other treatment of substrates such as semiconductor wafers.Example processes that may be performed on a substrate include, but arenot limited to, a plasma enhanced chemical vapor deposition (PECVD)process, a chemically enhanced plasma vapor deposition (CEPVD) process,a sputtering physical vapor deposition (PVD) process, an ionimplantation process, and/or other etch (e.g., chemical etch, plasmaetch, reactive ion etch, etc.), deposition, and cleaning processes. Asubstrate may be arranged on a substrate support, such as a pedestal, anelectrostatic chuck (ESC), etc. in a processing chamber of the substrateprocessing system. For example, during etching, a gas mixture includingone or more precursors is introduced into the processing chamber andplasma is struck to etch the substrate.

During process steps, process parameters (e.g., temperatures of variouscomponents of the system and the substrate, pressure within theprocessing chamber deposition rates, etch rates, power, etc.) may vary.These variations may have effects on the resulting substrates (e.g.,effects on critical dimensions of the substrates).

SUMMARY

A controller includes a memory that stores a first model correspondingto a first critical dimension of a substrate processed by a substrateprocessing system and a second model corresponding to a second criticaldimension of the substrate. The second model includes a predictedrelationship between the first critical dimension and the secondcritical dimension. A critical dimension prediction module calculates afirst prediction of the first critical dimension of the substrate usingthe first model, provides the first prediction of the first criticaldimension as an input to the second model, and calculates and outputs asecond prediction of the second critical dimension of the substrateusing the second model.

A method includes storing a first model corresponding to a firstcritical dimension of a substrate processed by a substrate processingsystem and a second model corresponding to a second critical dimensionof the substrate. The second model includes a predicted relationshipbetween the first critical dimension and the second critical dimension.The method further includes calculating a first prediction of the firstcritical dimension of the substrate using the first model, providing thefirst prediction of the first critical dimension as an input to thesecond model, and calculating and outputting a second prediction of thesecond critical dimension of the substrate using the second model.

Further areas of applicability of the present disclosure will becomeapparent from the detailed description, the claims and the drawings. Thedetailed description and specific examples are intended for purposes ofillustration only and are not intended to limit the scope of thedisclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure will become more fully understood from thedetailed description and the accompanying drawings, wherein:

FIG. 1 is a functional block diagram of an example substrate processingsystem according to the principles of the present disclosure;

FIG. 2A is a functional block diagram of an example system controlleraccording to the principles of the present disclosure;

FIG. 2B is a functional block diagram of a virtual metrology systemaccording to the principles of the present disclosure; and

FIG. 3 illustrates steps of a method for predicting critical dimensionsof a substrate using feedforward data according to the principles of thepresent disclosure.

In the drawings, reference numbers may be reused to identify similarand/or identical elements.

DETAILED DESCRIPTION

Critical dimensions of a semiconductor substrate (i.e., wafer) may beaffected by variations in process parameters, processing chambercharacteristics, etc. A critical dimension (CD) may refer to substratemetrology metrics such as line width, space width, gate length, holewidth, word line width, high aspect ratio hole width (e.g., at any depthor level in a respective hole such as a pillar), hole depth, trenchdepth, a resistance, etc. Some critical dimensions may be measuredsubsequent to processing. However, physical measurements of othercritical dimensions may be difficult and/or expensive, and, in somecases, may result in destruction of the substrate (e.g., by etching backthe substrate to take physical measurements). For example, physicalmeasurements may be difficult to obtain for three dimensionalstructures, such as in 3D NAND implementations. Accordingly, substrateprocessing systems may implement various systems and methods forpredicting/estimating critical dimensions of a processed substrate.

Virtual metrology systems and methods predict critical dimensions and/orother substrate processing system variables based on tool sensor data,measurement data, and/or other data. For example, virtual metrologysystems and methods may implement mathematical models that associateactual measured conditions (e.g., tool sensor data representing in situmeasurements taken using respective sensors) to other conditions withinthe processing chamber, characteristics of the substrate, etc. Themodels may be periodically updated according to sampled metrology dataand feedback data, including, but not limited to, physical measurementstaken after processing of the substrate is completed. In this manner,estimation of critical dimensions of substrates processed subsequent toupdating the models may be improved.

Virtual metrology systems and methods according to the principles of thepresent disclosure implement “within wafer” critical dimensionprediction of targeted structures using “within wafer” feedforward data.The feedforward data may correspond to virtual metrology modelpredictions performed for other structures on the same wafer. Forexample, a first set of critical dimensions of a processed substrate maybe predicted using a corresponding virtual metrology model. Thepredicted first set of critical dimensions may then be used as inputsfor predicting a second set of critical dimensions for the sameprocessed substrate. For example, the first set of critical dimensionsand the second set of critical dimensions may each be associated with asame feature of the processed substrate. Accordingly, the first set ofcritical dimensions may be indicative of the second set of criticaldimensions for a given substrate. Accordingly, a virtual metrology modelcorresponding to the second set of critical dimensions may be based inpart on a relationship between the first set of critical dimensions andthe second set of critical dimensions as described below in more detail.

Referring now to FIG. 1, an example substrate processing system 100 forperforming etching using RF plasma is shown. The substrate processingsystem 100 includes a processing chamber 102 that encloses othercomponents of the substrate processing system 100 and contains the RFplasma. The substrate processing chamber 102 includes an upper electrode104 and a substrate support, such as an electrostatic chuck (ESC) 106.During operation, a substrate 108 is arranged on the ESC 106.

For example only, the upper electrode 104 may include a showerhead 109that introduces and distributes process gases. The showerhead 109 mayinclude a stem portion including one end connected to a top surface ofthe processing chamber. A base portion is generally cylindrical andextends radially outwardly from an opposite end of the stem portion at alocation that is spaced from the top surface of the processing chamber.A substrate-facing surface or faceplate of the base portion of theshowerhead includes a plurality of holes through which process gas orpurge gas flows. Alternately, the upper electrode 104 may include aconducting plate and the process gases may be introduced in anothermanner.

The ESC 106 includes a conductive baseplate 110 that acts as a lowerelectrode. The baseplate 110 supports a heating plate 112, which maycorrespond to a ceramic multi-zone heating plate. A thermal resistancelayer 114 may be arranged between the heating plate 112 and thebaseplate 110. The baseplate 110 may include one or more coolantchannels 116 for flowing coolant through the baseplate 110.

An RF generating system 120 generates and outputs an RF voltage to oneof the upper electrode 104 and the lower electrode (e.g., the baseplate110 of the ESC 106). The other one of the upper electrode 104 and thebaseplate 110 may be DC grounded, AC grounded or floating. For exampleonly, the RF generating system 120 may include an RF voltage generator122 that generates the RF voltage that is fed by a matching anddistribution network 124 to the upper electrode 104 or the baseplate110. In other examples, the plasma may be generated inductively orremotely.

A gas delivery system 130 includes one or more gas sources 132-1, 132-2,. . . , and 132-N (collectively gas sources 132), where N is an integergreater than zero. The gas sources supply one or more precursors andmixtures thereof. The gas sources may also supply purge gas. Vaporizedprecursor may also be used. The gas sources 132 are connected by valves134-1, 134-2, . . . , and 134-N (collectively valves 134) and mass flowcontrollers 136-1, 136-2, . . . , and 136-N (collectively mass flowcontrollers 136) to a manifold 140. An output of the manifold 140 is fedto the processing chamber 102. For example only, the output of themanifold 140 is fed to the showerhead 109.

A temperature controller 142 may be connected to a plurality of thermalcontrol elements (TCEs) 144 arranged in the heating plate 112. Forexample, the TCEs 144 may include, but are not limited to, respectivemacro TCEs corresponding to each zone in a multi-zone heating plateand/or an array of micro TCEs disposed across multiple zones of amulti-zone heating plate as described in more detail in FIGS. 2A and 2B.The temperature controller 142 may be used to control the plurality ofTCEs 144 to control a temperature of the ESC 106 and the substrate 108.

The temperature controller 142 may communicate with a coolant assembly146 to control coolant flow through the channels 116. For example, thecoolant assembly 146 may include a coolant pump and reservoir. Thetemperature controller 142 operates the coolant assembly 146 toselectively flow the coolant through the channels 116 to cool the ESC106.

A valve 150 and pump 152 may be used to evacuate reactants from theprocessing chamber 102. A system controller 160 may be used to controlcomponents of the substrate processing system 100. A robot 170 may beused to deliver substrates onto, and remove substrates from, the ESC106. For example, the robot 170 may transfer substrates between the ESC106 and a load lock 172. Although shown as separate controllers, thetemperature controller 142 may be implemented within the systemcontroller 160. The system controller 160 or a separately locatedvirtual metrology controller may implement the virtual metrology systemsand methods according to the principles of the present disclosure.

Referring now to FIG. 2A, an example system controller 200 includes datacollection module 204, critical dimension (CD) prediction module 208,processing control module 212, and memory 216. For example, memory 216may correspond to non-volatile memory, such as non-volatilesemiconductor memory. Memory 216 stores one or more virtual metrologymodels configured to estimate respective critical dimensions of asubstrate based on a plurality of inputs during and/or subsequent toprocessing.

For example, CD prediction module 208 may be configured to implement themodels during and/or subsequent to processing to predict variouscritical dimensions of the substrate. The CD prediction module 208executes the models using inputs received from data collection module204. For example, the data collection module 204 may receive inputs 220including, but not limited to, physical measurements of the substratetaken prior to processing (e.g., as input by a user), sensor datacorresponding to sensor measurements (e.g., gas flow rates, temperature,pressure, RF power, etc.) taken from within the processing chamber 102during processing, etc. The data collection module 204 may alsocalculate and/or predict various processing parameters using the inputs220 (e.g., tool sensor data), inputs from the processing control module212, etc. For example, the processing control module 212 may beconfigured to control various parameters associated with substrateprocessing, including, but not limited to, gas flow rates, powerprovided to components of the substrate processing system 100,temperatures, etc. The processing control module 212 may providefeedback indicative of the controlled parameters to the data collectionmodule 204.

Accordingly, the CD prediction module 208 retrieves the models stored inthe memory 216 and implements the models according to inputs from thedata collection module 204, the processing control module 212, etc. Theinputs to the models correspond to physical measurements of thesubstrate being processed, conditions within the processing chamber 102,control parameters associated with the substrate processing system 100,and/or any data incorporated into the models for calculating thecritical dimensions of the substrate. The CD prediction module 208according to the principles of the present disclosure is configured tofurther provide results of a first model (i.e., corresponding to a firstset of predicted critical dimensions) as inputs to a second model. Inother words, results of the second model (i.e., corresponding to asecond set of predicted critical dimensions) are calculated based inpart on the results of the first model.

For example only, the second model may be constructed according to apredicted relationship between a first critical dimension (e.g.,corresponding to the first set of critical dimensions) and a secondcritical dimension (e.g., corresponding to the second set of criticaldimensions). The predicted relationship may be calculated according todata collected for a plurality of processed substrates. Criticaldimensions of the processed substrates may be measured to determine arelationship between a first critical dimension CDx and a secondcritical dimension CDy of the same substrate, where CDx may correspondto a critical dimension that may be accurately physically measuredsubsequent to processing while CDy may correspond to a criticaldimension that may not be accurately physically measured subsequent toprocessing.

In one example, the first critical dimension CDx may correspond to acritical dimension at tops of high aspect ratio structures (e.g. a widthbetween tops of high aspect ratio pillars), which may be physicallymeasurable subsequent to processing. Conversely, the second criticaldimension CDy may correspond to a critical dimension at bottoms of thehigh aspect ratio structures (e.g., a width between bottoms of the highaspect ratio pillars), which may be difficult to physically measuresubsequent to processing. Accordingly, the predicted relationshipbetween the first critical dimension CDx and the second criticaldimension CDy may be calculated based on processing results of aplurality of substrates by measuring the first critical dimension CDxand the second critical dimension CDy. For example only, the substratemay be etched back to measure the second critical dimension CDy toobtain the measurements for calculating the predicted relationship. Thepredicted relationship may correspond to an approximately linear ornon-linear slope, a derivative, a high-order equation, etc. In anotherexample, the first critical dimension CDx corresponds to a height of ahighest high aspect ratio pillar and the second critical dimension CDycorresponds to a height of a lowest high aspect ratio pillar.

In this manner, the CD prediction module 208 calculates a first set ofcritical dimensions (e.g., CDx-1, CDx-2, . . . , CDx-N) of a substrateusing the first model. The calculated first set of critical dimensionsare input to the second model, and the CD prediction module 208calculates a second set of critical dimensions (CDy-1, CDy-2, . . . ,CDy-N) of the same substrate using the second model and the first set ofcritical dimensions. The second model, as previously constructed andstored in the memory 216, incorporates the predicted relationshipbetween the first set of critical dimensions and the second set ofcritical dimensions.

Physical measurements of the first set of critical dimensions takensubsequent to the processing of the substrate may be compared with thecalculated results of the first model to update the second model. Forexample, since the first set of critical dimensions are indicative ofthe second set of critical dimensions according to the predictedrelationship, it may be presumed that differences (e.g., an offset ordelta) between the predictions and actual measurements of the first setof critical dimensions may be indicative of differences between thepredictions and actual measurements of the second set of criticaldimensions. Accordingly, the differences between the predictions andactual measurements of the first set of critical dimensions may beprovided as inputs to the second model to obtain a more accurateprediction of the second set of critical dimensions.

In other examples, the second model may further incorporate predictedrelationships between critical dimensions within the first set ofcritical dimensions. For example, a difference DX1 between CDx-1 andCDx-2 may be indicative of a difference DXY between CDx-1 and CDy-1. Assuch, a relationship between CDx-1 and CDy-1 may be further definedaccording to a ratio of DX1 to DXY. In other words, a change in DX1 dueto a change in either of CDx-1 or CDx-2 may be used to calculate acorresponding change in CDy-1 by assuming the change in CDy-1 conformsto the ratio DX1 to DXY.

Referring now to FIG. 2B, in addition to and/or instead of a localsystem controller 160/200 as described in FIGS. 1 and 2A, a virtualmetrology system 224 may include a remotely located virtual metrologycontroller 228 implementing the virtual metrology systems and methodsdescribed herein. For example, a fabrication facility 232 may includeone or more substrate processing systems 236 (e.g., corresponding to thesubstrate processing system 100 described in FIG. 1). The substrateprocessing system 236 communicates with the remotely located virtualmetrology controller 228 via a network (e.g., a wired or wirelessnetwork) 240. A virtual metrology server 244 may implement the virtualmetrology controller 228. Although shown outside of the fabricationfacility 232, the virtual metrology server 244 may be located within thefacility 232 in some examples.

The virtual metrology controller 228 may include components functionallyanalogous to the memory 216, the CD prediction module 208, the datacollection module 204, the processing control module 212, etc. asdescribed in FIG. 2A. Accordingly, the virtual metrology controller 228is configured to calculate predictions of critical dimensions asdescribed above with respect to FIG. 2A. The virtual metrologycontroller 224 may receive metrology data, including historical data,from the substrate processing system 236 and/or directly from othersources 248. The other sources 248 may include, but are not limited to,stored historical data, other fabrication facilities, user inputs, etc.A user may access the virtual metrology controller 228 via a clientdevice 252 (e.g., a personal computer, laptop, or other computingdevice).

Referring now to FIG. 3, an example method 300 for predicting criticaldimensions of a substrate using feedforward data according to theprinciples of the present disclosure begins at 304. At 308, a predictedrelationship between a first critical dimension and a second criticaldimension is calculated. For example, the predicted relationship may becalculated according to physical measurements taken of the firstcritical dimension and the second critical dimension for a plurality ofsubstrates. At 312, a model is constructed based on the predictedrelationship. For example, the model corresponds to a virtual metrologymodel stored in memory and configured to output a prediction of thesecond critical dimension using a prediction of the first criticaldimension as one of a plurality of inputs.

At 316, the method 300 (e.g., the CD prediction module 208) calculates aprediction of the first critical dimension. For example, the method 300receives one or more inputs related to the processing of the substrateand implements a virtual metrology model associated with the firstcritical dimension. At 320, the method 300 (e.g., the CD predictionmodule 208) calculates, using the model based on the predictedrelationship between the first critical dimension and the secondcritical dimension, a prediction of the second critical dimension. Forexample, the CD prediction module 208 uses the prediction of the firstcritical dimension as one of a plurality of inputs to the model. Themethod 300 ends at 324.

The foregoing description is merely illustrative in nature and is in noway intended to limit the disclosure, its application, or uses. Thebroad teachings of the disclosure can be implemented in a variety offorms. Therefore, while this disclosure includes particular examples,the true scope of the disclosure should not be so limited since othermodifications will become apparent upon a study of the drawings, thespecification, and the following claims. It should be understood thatone or more steps within a method may be executed in different order (orconcurrently) without altering the principles of the present disclosure.Further, although each of the embodiments is described above as havingcertain features, any one or more of those features described withrespect to any embodiment of the disclosure can be implemented in and/orcombined with features of any of the other embodiments, even if thatcombination is not explicitly described. In other words, the describedembodiments are not mutually exclusive, and permutations of one or moreembodiments with one another remain within the scope of this disclosure.

Spatial and functional relationships between elements (for example,between modules, circuit elements, semiconductor layers, etc.) aredescribed using various terms, including “connected,” “engaged,”“coupled,” “adjacent,” “next to,” “on top of,” “above,” “below,” and“disposed.” Unless explicitly described as being “direct,” when arelationship between first and second elements is described in the abovedisclosure, that relationship can be a direct relationship where noother intervening elements are present between the first and secondelements, but can also be an indirect relationship where one or moreintervening elements are present (either spatially or functionally)between the first and second elements. As used herein, the phrase atleast one of A, B, and C should be construed to mean a logical (A OR BOR C), using a non-exclusive logical OR, and should not be construed tomean “at least one of A, at least one of B, and at least one of C.”

In some implementations, a controller is part of a system, which may bepart of the above-described examples. Such systems can comprisesemiconductor processing equipment, including a processing tool ortools, chamber or chambers, a platform or platforms for processing,and/or specific processing components (a wafer pedestal, a gas flowsystem, etc.). These systems may be integrated with electronics forcontrolling their operation before, during, and after processing of asemiconductor wafer or substrate. The electronics may be referred to asthe “controller,” which may control various components or subparts ofthe system or systems. The controller, depending on the processingrequirements and/or the type of system, may be programmed to control anyof the processes disclosed herein, including the delivery of processinggases, temperature settings (e.g., heating and/or cooling), pressuresettings, vacuum settings, power settings, radio frequency (RF)generator settings, RF matching circuit settings, frequency settings,flow rate settings, fluid delivery settings, positional and operationsettings, wafer transfers into and out of a tool and other transfertools and/or load locks connected to or interfaced with a specificsystem.

Broadly speaking, the controller may be defined as electronics havingvarious integrated circuits, logic, memory, and/or software that receiveinstructions, issue instructions, control operation, enable cleaningoperations, enable endpoint measurements, and the like. The integratedcircuits may include chips in the form of firmware that store programinstructions, digital signal processors (DSPs), chips defined asapplication specific integrated circuits (ASICs), and/or one or moremicroprocessors, or microcontrollers that execute program instructions(e.g., software). Program instructions may be instructions communicatedto the controller in the form of various individual settings (or programfiles), defining operational parameters for carrying out a particularprocess on or for a semiconductor wafer or to a system. The operationalparameters may, in some embodiments, be part of a recipe defined byprocess engineers to accomplish one or more processing steps during thefabrication of one or more layers, materials, metals, oxides, silicon,silicon dioxide, surfaces, circuits, and/or dies of a wafer.

The controller, in some implementations, may be a part of or coupled toa computer that is integrated with the system, coupled to the system,otherwise networked to the system, or a combination thereof. Forexample, the controller may be in the “cloud” or all or a part of a fabhost computer system, which can allow for remote access of the waferprocessing. The computer may enable remote access to the system tomonitor current progress of fabrication operations, examine a history ofpast fabrication operations, examine trends or performance metrics froma plurality of fabrication operations, to change parameters of currentprocessing, to set processing steps to follow a current processing, orto start a new process. In some examples, a remote computer (e.g. aserver) can provide process recipes to a system over a network, whichmay include a local network or the Internet. The remote computer mayinclude a user interface that enables entry or programming of parametersand/or settings, which are then communicated to the system from theremote computer. In some examples, the controller receives instructionsin the form of data, which specify parameters for each of the processingsteps to be performed during one or more operations. It should beunderstood that the parameters may be specific to the type of process tobe performed and the type of tool that the controller is configured tointerface with or control. Thus as described above, the controller maybe distributed, such as by comprising one or more discrete controllersthat are networked together and working towards a common purpose, suchas the processes and controls described herein. An example of adistributed controller for such purposes would be one or more integratedcircuits on a chamber in communication with one or more integratedcircuits located remotely (such as at the platform level or as part of aremote computer) that combine to control a process on the chamber.

Without limitation, example systems may include a plasma etch chamber ormodule, a deposition chamber or module, a spin-rinse chamber or module,a metal plating chamber or module, a clean chamber or module, a beveledge etch chamber or module, a physical vapor deposition (PVD) chamberor module, a chemical vapor deposition (CVD) chamber or module, anatomic layer deposition (ALD) chamber or module, an atomic layer etch(ALE) chamber or module, an ion implantation chamber or module, a trackchamber or module, and any other semiconductor processing systems thatmay be associated or used in the fabrication and/or manufacturing ofsemiconductor wafers.

As noted above, depending on the process step or steps to be performedby the tool, the controller might communicate with one or more of othertool circuits or modules, other tool components, cluster tools, othertool interfaces, adjacent tools, neighboring tools, tools locatedthroughout a factory, a main computer, another controller, or tools usedin material transport that bring containers of wafers to and from toollocations and/or load ports in a semiconductor manufacturing factory.

What is claimed is:
 1. A controller, comprising: a memory that stores(i) a first model corresponding to a first critical dimension of asubstrate processed by a substrate processing system and (ii) a secondmodel corresponding to a second critical dimension of the substrate,wherein the second model includes a predicted relationship between thefirst critical dimension and the second critical dimension; and acritical dimension prediction module that (i) calculates a firstprediction of the first critical dimension of the substrate using thefirst model, (ii) provides the first prediction of the first criticaldimension as an input to the second model, and (iii) calculates andoutputs a second prediction of the second critical dimension of thesubstrate using the second model.
 2. The controller of claim 1, whereinat least one of the first model and the second model corresponds to avirtual metrology model.
 3. The controller of claim 1, wherein the firstcritical dimension and the second critical dimension are associated witha same feature of the substrate.
 4. The controller of claim 1, whereinthe first critical dimension and the second critical dimensioncorrespond to a height of a high aspect ratio structure on thesubstrate.
 5. The controller of claim 1, wherein the first criticaldimension and the second critical dimension correspond to a widthbetween high aspect ratio structures on the substrate.
 6. The controllerof claim 1, wherein the critical dimension prediction module updates thesecond model based on a comparison between the first prediction of thefirst critical dimension and a measurement of the first criticaldimension.
 7. The controller of claim 1, wherein the predictedrelationship is based on respective measurements of the first criticaldimension and the second critical dimension on a plurality ofsubstrates.
 8. The controller of claim 1, wherein the predictedrelationship is based on a ratio of (i) a difference between the firstcritical dimension and a third critical dimension to (ii) a differencebetween the first critical dimension and the second critical dimension.9. The controller of claim 1, wherein the predicted relationshipcorresponds to a linear or a non-linear relationship between the firstcritical dimension and the second critical dimension.
 10. A system,comprising: the controller of claim 1; and the substrate processingsystem, wherein the controller is remotely located from the substrateprocessing system.
 11. A method, comprising: storing (i) a first modelcorresponding to a first critical dimension of a substrate processed bya substrate processing system and (ii) a second model corresponding to asecond critical dimension of the substrate, wherein the second modelincludes a predicted relationship between the first critical dimensionand the second critical dimension; calculating a first prediction of thefirst critical dimension of the substrate using the first model;providing the first prediction of the first critical dimension as aninput to the second model; and calculating and outputting a secondprediction of the second critical dimension of the substrate using thesecond model.
 12. The method of claim 11, wherein at least one of thefirst model and the second model corresponds to a virtual metrologymodel.
 13. The method of claim 11, wherein the first critical dimensionand the second critical dimension are associated with a same feature ofthe substrate.
 14. The method of claim 11, wherein the first criticaldimension and the second critical dimension correspond to a height of ahigh aspect ratio structure on the substrate.
 15. The method of claim11, wherein the first critical dimension and the second criticaldimension correspond to a width between high aspect ratio structures onthe substrate.
 16. The method of claim 11, further comprising updatingthe second model based on a comparison between the first prediction ofthe first critical dimension and a measurement of the first criticaldimension.
 17. The method of claim 11, wherein the predictedrelationship is based on respective measurements of the first criticaldimension and the second critical dimension on a plurality ofsubstrates.
 18. The method of claim 11, wherein the predictedrelationship is based on a ratio of (i) a difference between the firstcritical dimension and a third critical dimension to (ii) a differencebetween the first critical dimension and the second critical dimension.19. The method of claim 11, wherein the predicted relationshipcorresponds to a linear or a non-linear relationship between the firstcritical dimension and the second critical dimension.